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1898First Tradable Session 3A signal becomes public at Thursday 6:00pm ET. A daily strategy may only trade after all input data are known. What is the earliest tradable session?统计中等数值题未尝试免费2866Markov Bound for Daily SlippageA nonnegative slippage random variable L has mean E[L]=2 basis points. Give the best Markov upper bound you can on P(L\ge 10).概率简单derivation未尝试面试订阅2867A Generalized Markov Bound from the Fourth MomentSuppose X is any random variable with E[X 4]=81. Use Markov's inequality on a suitable nonnegative variable to bound P(|X|\ge 6).概率中等derivation未尝试面试订阅2868Recovering Chebyshev from MarkovDerive Chebyshev's inequality from Markov's inequality. In other words, show that for any random variable X with mean and variance 2, \[ P(|X- |\ge a)\le 2 a 2 . \]概率简单derivation未尝试面试订阅2869Chebyshev for a Monte Carlo MeanAn unbiased Monte Carlo estimator averages n=100 i.i.d. draws with variance 9. Use Chebyshev's inequality to bound the probability that the sample mean deviates from its target by at least 0.5.概率简单derivation未尝试面试订阅2870Which Bound Is Better Here?A nonnegative random variable X satisfies E[X]=1 and Var (X)=4. Compare the Markov and Chebyshev upper bounds on P(X\ge 5), and say which one is tighter.概率中等derivation未尝试面试订阅2873Hoeffding Versus Chebyshev for 65 Heads in 100 TossesA fair coin is tossed 100 times. Compare the Hoeffding and Chebyshev upper bounds on the event that the fraction of heads is at least 0.65.概率中等derivation未尝试面试订阅2890Best Available Bound for a Bounded MeanYou average n=200 independent observations in [0,1]. Compare the Chebyshev and Hoeffding upper bounds on \[ P ( X-E[ X]\ge 0.1 ). \] Use the worst-case variance for the Chebyshev side.概率中等derivation未尝试面试订阅5906How Many Bets Until Loss Is UnlikelyA Kelly bettor on an even-money coin with p=0.6 stakes the optimal fraction f *=0.2 each round. The per-round log-return is +\ln 1.2 with probability 0.6 and \ln 0.8 with probability 0.4, with mean G\approx0.0201 and variance v\approx0.0395. Using Chebyshev's inequality, find a number of rounds n after which the probability of ending below the starting wealth is at most 5\%.概率困难数值题未尝试面试订阅